Higher June 2024 Paper 4 Q14

OCRHigherCurrent spec6 marksAdvanced Trigonometry Circles & Sectors

14 The diagram shows a circle, centre O and radius 52 cm.

Circle centre O, radius 52 cm, with chord AB at the top; the segment above AB is shaded; a tangent touches the circle at T at the bottom; the perpendicular distance from the chord to the tangent is 72 cm

Not to scale

AB is a chord of the circle.
The line through T is a tangent to the circle.

The chord is parallel to the tangent.
The perpendicular distance between the chord and the tangent is 72 cm.

Calculate the area of the shaded segment.
You must show your working. [6]

Mark scheme

Answer	Marks	Part marks and guidance
2219.9… or 2220 with correct working	6		“Correct working” requires evidence of at least M2 or M1 M1 Accept answers in the range 2217 to 2226. Accept any correct method.
		M2 for \(\cos [\ldots] = \frac{72 - 52}{52}\) oe or M1 for 72 – 52 or 20 and	M2 implied by 67.38… or 134.76… rot to at least an integer e.g. 67, 67.3, 67.4, 67.38, 134, equivalent method includes \(\cos [\ldots] = \frac{52^2 + 52^2 - 96^2}{2 \times 52 \times 52}\) condone with 72 instead of 96
		M2 for \(\frac{1}{2} \times 52 \times 52 \times \sin\) their 134.76… oe or M1 for \(\sqrt{52^2 - (\textit{their } 20)^2}\) oe or 48 and M1 for \(\frac{1}{2} \times\) their96 × their20 oe and	M2 for finding the area of the triangle OAB implied by 960 their 134.76… must come from a correct attempt to find angle AOB and condone using AB = 72 for M2 and M1 including \(\sqrt{52^2 - 36^2}\) for M1
		M1 for \(\frac{\textit{their angle AOB}}{360} \times \pi \times 52^2\) or B1 for \(\pi \times 52^2\)	M1 implied by 3176 to 3186 or B1 implied by 2704\(\pi\), 8490 to 8496
		If 0, 1 or 2 scored award SC3 for the correct answer with insufficient working If 0 or 1 scored award SC2 for 3179.918… or 5314.718.. rot to at least an integer with insufficient working	alternative method using major sector: M2 for angle as before implied by 225, 225.2, 225.23 or 225.24 M2 for area of triangle as before or M1 for the third side e.g. 48 M1 for \(\left[\frac{\textit{their reflex angle AOB}}{360}\right] \times \pi \times 52^2\) or 5314.718..